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Numerical methods and modelling for engineering
Khoury R., Harder D., Springer International Publishing, New York, NY, 2016. 332 pp. Type: Book (978-3-319211-75-6)
Date Reviewed: Feb 23 2017

I always appreciated numerical methods courses, possibly because I found the field a simpler and more versatile alternative to learning the analytic solutions across many application domains. Although this book is intended as a textbook for students, with exercises and questions at the end of each chapter, it is equally suitable for practicing engineers or computer scientists who find themselves using computers to solve numerical problems. Given that many algorithms are named after their founder, I apologize if this review sounds too much like a who’s who of the numerical methods field.

The book has an excellent progression in complexity, with the initial chapters requiring very little prerequisite knowledge and the later ones assuming some understanding of the mathematical domains being approximated. Starting with a brief coverage on how precision and accuracy feed into error analysis, the authors then detail floating-point precision and the limitations of hardware implementations thereof. Chapter 3 discusses iteration and the importance of stopping conditions due to either convergence or reaching some maximum iteration count.

Introductions aside, chapter 4 covers linear algebra, providing explanations, examples, and pseudocode for PLU and Cholesky decomposition, and both the Jacobi and the Gaus-Seidel methods for iterative solutions to matrix systems where a good initial guess is available. Chapter 5 covers the Taylor series as well as its use in error analysis. Subsequent chapters then use the Taylor series to estimate the accuracy of the various methods and algorithms. The following chapter is a significantly more detailed coverage of interpolation, regression, and extrapolation. For interpolation, the authors describe the Vandermonde method and both Lagrange and Newton polynomials for curve fitting, with the Vandermonde method featured again in the regression section alongside least squares.

Chapter 7 describes the concept of bracketing algorithms and the binary search method, which are put to use in the following chapter on root-finding. Here, the available algorithms are numerous and the authors describe the bisection and false position closed interval methods, and also the fixed-point iteration, Newton’s, secant, and Muller’s open interval methods. A multidimensional example of the Newton method together with pseudocode is provided. Root-finding leads naturally to functional optimization where we want to find either a maximum or minimum of a function over some interval. Hence, chapter 9 is devoted to the golden mean algorithm, quadratic optimization, gradient descent, brute force random search, and details on simulated annealing.

Chapter 10 is dedicated to differentiation and looks into finite difference techniques for approximating the derivative of some unknown function. Richardson extrapolation is described as a method for obtaining increased accuracy for your derivative approximation without the risk of numerical problems caused by subtractive cancellation. Chapter 11 covers integration. Moving up the ranks of complexity, and also asymptotic accuracy, the chapter details the trapezoid rule, Romberg integration, both the 1/3 and 3/8 Simpson’s rules, and ends with Gaussian quadrature.

The final two chapters cover initial value and boundary value problems, respectively. It is only these two chapters that might be beyond university entry level. For the former, the chapter looks at Euler’s (both forward and backward), Huen’s, and the fourth-order Runge-Kutta methods. Using the matrix methods from chapter 4, the aforementioned methods are applied to both systems of initial value problems as well as higher-order ordinary differential equations. The chapter on boundary value problems (where we know the value of the function at its edges, have information on its derivative, and need to evaluate its behavior between the known boundaries) covers the shooting method as well as finite difference techniques applied to both single and multidimensional functions.

Two appendices are provided. The first is a brief description of pseudocode as used within the book, while the latter contains the answers to the numerous problems and exercises at the end of each chapter. Overall, I found this book very easy to read and follow, with chapters flowing naturally on from each other. Although no book could ever claim to be exhaustive on the topic matter, this introductory text on numeric methods does provide just the right level of coverage required of a generalist.

Reviewer:  Bernard Kuc Review #: CR145076 (1705-0245)
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